Question

How do you determine the concavity of a quadratic function?

Answer 1

Use the concept of the second derivative:

The quadratic function can be written as: $f\left(x\right)=a{x}^2+bx+c$

where, $a\ne 0$

Now, differentiate the function:

$$\begin{gathered} f\text{'}\left(x\right)=\frac{d}{dx}\left(a{x}^2+bx+c\right) \\ \Rightarrow =2ax+b \end{gathered}$$

Again differentiate the function:

,$$\begin{gathered} f\text{'}\text{'}\left(x\right)=\frac{d}{dx}\left(2ax+b\right) \\ \Rightarrow =2a \end{gathered}$$

Now, if $a>0$ then $f\text{'}\text{'}\left(x\right)$ is positive and the graph of a quadratic function is concave up.

and, if $a<0$ then $f\text{'}\text{'}\left(x\right)$ is negative and the graph of a quadratic function is concave down.

Hence, to determine the concavity of quadratic function, double derivate the function and check the sign of $a$. If the sign is positive then it is concave upwards and if it is negative then is concave downwards.